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Full text of "Treatise On Analysis Vol-Ii"

126       XIII   INTEGRATION

Since the sequence (gn) is increasing and \ gn d[i^ I * gn dpi < + oo, the result
follows from (13.8.1).

(13.8.3)   Let (fn) be a sequence of integr able functions, and suppose that there

/*
hdjn< 4-00 and  h ^fn :g h almost

every where, for alln. Then the functions lim inf/, andlim sup/, are integrable,

-+ oo                        n~*oo

and we have

(13.8.3.1)      Aiminf/B) d\i  lim inf   / d/j,

J   \   n~*oo         /                n-*oo     J

^ lim sup Ifndfj,^    (lim sup/J d\i.

n~*oo      J                  J   \     n-*-oo         /

It is enough to prove that lim inffn is integrable and that the left-hand

n--oo

inequality in (13.8.3.1) is valid; the other assertions then follow by replacing
fn by  /, . Let gn = inf/n+ p . It follows from (1 3.8.2) applied to the sequence

p^O

(fn+p)Po and from ^e hypothesis fn+p ^  h, that gn is integrable, and

clearly f gn d/j,  (fn+p d\i for all p ^ 0, so that f gn d^ g inf (fn+p d\i. The

J                   J                                                                    J                   pQJ

sequence (gn) is increasing ; we have gn^h for all n, and by definition (1 2.7.1 0),

lim infyj, = sup gn . Hence it follows from (13.8.1) that lim inf/w is integrable

n-oo                 n                                                                                             n-*oo

and that

(lim inf/J d/j, = sup    gn dp ^ sup ( inf   fn+p d/A

J   \    n~*oo        /               n     J                    n     \p0 J               /

= lim inf   / dfi.

w--oo     J

(13.8.4)       (Dominated convergence theorem)   Let (fn) be a sequence of
integrable functions ; suppose that Iimfn(x) =f(x) exists almost everywhere

n-* oo

and that there exists a function g ^ 0 such that j   g d/j, < +00 and \fn\ ^ g
almost everywhere, for alln. Then f is integrable and

(13.8.4.1)                    f/(x) dftx) .= lim (/(*) d/i(x).

J                               B-+00-J

This follows from (13.8.3). In this situation the two extreme terms of
(13.8.3.1) are equal, whence the result.f:X-+Htobe integrable it is necessary and